On the other hand, in terms of geometric interpretation, a modular form can be regarded as a section of a certain sheaf of differential forms on the open modular curve on a congruence subgroup \(\Gamma \subset SL_2(\mathbb {Z}).\) 
By noting that the singular cohomology of the open modular curve is given by the group cohomology \(H^*(\Gamma , W)\)  the comparison of de Rham and singular cohomology can give an Eichler-isomorphism.
Matsushima and Murakami [1]} extended the results to show that the space of automorphic forms on a symmetric space \(M\)  is isomorphic to \(H^*(M,S)\)  for a certain locally constant sheaf \(S\)  over \(M.\)  The cohomology of Hilbert surfaces in terms of Hilbert cusp forms has been studied by many researchers including [2]}, [3]}. Relating the critical values of \(L\) -functions of Hilbert cusp forms and cohomology was first studied by Yoshida [3]}.
Following the work by Knopp [5]} and Kohnen-Zagier [6]}, which provide us
the considerable new lights on Eichler-Shimura isomorphisms such as rational structures of elliptic modular forms,
we are able to
associate the space of Hilbert modular forms over the totally real number fields to the parabolic cohomology group in terms of the period polynomial by taking anti-derivative of Hilbert modular form [7]}.
